Chapter 5, Problem 14CT

(a)

To determine

To graph: the inequality that represents the numbers of trophies and medals it can buy and interpret a solution of the inequality.

Answer to Problem 14CT

The linear inequality is $12x+3y\le 60$ .

Explanation of Solution

Given:

The person wants to spend at most $60$ dollar on trophies and medals. The trophy is $12$ dollar each and the medal is $3$ dollar each.

Concept used:

Graphing the inequalities, it will graph the ordinary linear functions just like it is done before. The difference is that a solution to the inequality is not the drawn line but the are of the coordinate plane that satisfy the inequality.

The boundary line is dashed for $<\text{or >}$ and solid line for $\le \text{or }\ge$ .

The common area is the solution of inequality.

Calculation:

The person wants to spend at most $60$ dollar on trophies and medals. The trophy is $12$ dollar each and the medal is $3$ dollar each.

Write the inequality and draw the graph.

If it wants purchase at least $6$ items, write and draw the system of linear inequalities, determine how many of each item it can be purchased.

Let $x$ be the number of trophies and $y$ be the number of medals. Then the inequalities that models the purchase with at most $60$ dollar is:

  $12x+3y\le 60$ .

Since, total amount is at most $60$ dollars.

The linear inequality is $12x+3y\le 60$ .

Draw the graph of $12x+3y\le 60$ as show below. If only one item is purchase, either $20$ trophies or $5$ medal can be purchase as it is clear from the graph.

The graph of equation $y\le -4x+20$ :

Hence, the linear inequality is $12x+3y\le 60$ .

(b)

To determine

To graph: the system that represents the situation and find the number of item that person can buy.

Answer to Problem 14CT

The graph of system of linear inequalities is drawn.

Explanation of Solution

Given:

The person wants to spend at most $60$ dollar on trophies and medals. The trophy is $12$ dollar each and the medal is $3$ dollar each.

Concept used:

Graphing the inequalities, it will graph the ordinary linear functions just like it is done before. The difference is that a solution to the inequality is not the drawn line but the are of the coordinate plane that satisfy the inequality.

The boundary line is dashed for $<\text{or >}$ and solid line for $\le \text{or }\ge$ .

The common area is the solution of inequality.

Calculation:

If it purchased at least $6$ items then the system of inequalities is:

  $\begin{array}{l}12x+3y\le 60.\\ x+y\ge 6.\end{array}$

Graph the inequalities $x+y\ge 6\text{ and }12x+3y\le 60$ as shown in the below figure. The intersection of blue and red colours represents the solutions set of given system of inequalities. The ordered pair $\left(x,y\right)=\left(2,10\right)$ is in the solution set. It means it can buy $2$ trophies and $10$ medals.

The graph of inequalities $y\le -4x+20,y\ge -x+6$ is:

Hence, the graph of system of linear inequalities is drawn.